To find the equations of the circles that pass through the origin and cut off specific intercepts from the axes, we can follow these steps:
### Part (i): Intercepts of 3 and 4
1. **Identify the intercepts**: The intercept on the x-axis is 3, and the intercept on the y-axis is 4. Therefore, the points where the circle intersects the axes are (3, 0) and (0, 4).
2. **Determine the center of the circle**: The center of the circle will be at (h, k), where:
- The x-intercept is 3, so the distance from the center to the x-axis is \( |h| = \frac{3}{2} \).
- The y-intercept is 4, so the distance from the center to the y-axis is \( |k| = 2 \).
Since the circle passes through the origin (0, 0), we can assume the center is at \( (h, k) = \left(\frac{3}{2}, 2\right) \).
3. **Write the general equation of the circle**: The general equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = R^2
\]
Substituting \( h = \frac{3}{2} \) and \( k = 2 \):
\[
(x - \frac{3}{2})^2 + (y - 2)^2 = R^2
\]
4. **Find the radius \( R \)**: Since the circle passes through the origin (0, 0), we can substitute these coordinates into the equation to find \( R \):
\[
(0 - \frac{3}{2})^2 + (0 - 2)^2 = R^2
\]
\[
\frac{9}{4} + 4 = R^2
\]
\[
R^2 = \frac{9}{4} + \frac{16}{4} = \frac{25}{4}
\]
5. **Final equation of the circle**: Substitute \( R^2 \) back into the equation:
\[
(x - \frac{3}{2})^2 + (y - 2)^2 = \frac{25}{4}
\]
### Part (ii): Intercepts of 2a and 2b
1. **Identify the intercepts**: The intercepts are given as \( 2a \) on the x-axis and \( 2b \) on the y-axis. Thus, the points are \( (2a, 0) \) and \( (0, 2b) \).
2. **Determine the center of the circle**: The center will be at \( (h, k) \), where:
- The x-intercept is \( 2a \), so \( |h| = a \).
- The y-intercept is \( 2b \), so \( |k| = b \).
Since the circle passes through the origin, we can assume \( (h, k) = (a, b) \).
3. **Write the general equation of the circle**:
\[
(x - a)^2 + (y - b)^2 = R^2
\]
4. **Find the radius \( R \)**: Substitute the origin into the equation:
\[
(0 - a)^2 + (0 - b)^2 = R^2
\]
\[
a^2 + b^2 = R^2
\]
5. **Final equation of the circle**: Substitute \( R^2 \) back into the equation:
\[
(x - a)^2 + (y - b)^2 = a^2 + b^2
\]
### Summary of Solutions:
- For intercepts of 3 and 4:
\[
(x - \frac{3}{2})^2 + (y - 2)^2 = \frac{25}{4}
\]
- For intercepts of \( 2a \) and \( 2b \):
\[
(x - a)^2 + (y - b)^2 = a^2 + b^2
\]
